Unless you changed numbers I wouldn't think changing to the new SIM card would change the caller ID.
I had to have this done last year when I joined Visible, my ID showed cellular caller or something like that. It was explained to me that your number goes through database when you dial out and you can have it say whatever you want it to. So, you need to contact customer service and hopefully you get someone who knows what they are doing. Ask them to change your caller ID name, if they say they can't or don't know how ask for a supervisor and ask them, keep asking for someone until you find someone that knows how to do it. They will put in a ticket to whoever runs the database whether Visible or a third party and you can have it changed. I got lucky and the first person I chatted with got it done for me and the next day my caller ID was showing my full name.
It is my suggestion that it may be quicker to use Facebook messenger to get a hold of customer care. You could also try Twitter. You can find links in the bottom right hand corner of this page, should be on the help page also. I personally don't patience to wait for someone on chat for over an hour or more.
